mathAnalysis4-07-draft

mathAnalysis4-07-draft - P r e l i m i n a r y d r a f t o...

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Unformatted text preview: P r e l i m i n a r y d r a f t o n l y : p l e a s e c h e c k f o r fi n a l v e r s i o n ARE211, Fall 2007 LECTURE #4: TUE, SEP 11, 2007 PRINT DATE: AUGUST 21, 2007 (ANALYSIS4) Contents 1. Analysis (cont) 1 1.8. Topology of R n 1 1.8.1. Open Sets 1 1.8.2. Interior of a Set 3 1.8.3. Closed Sets 4 1.8.4. Accumulation Point 4 1. Analysis (cont) 1.8. Topology of R n 1.8.1. Open Sets. Definition: Given a universe X , a metric d and x ∈ X , the set B d ( x,epsilon1 | X ) = { y ∈ X : d ( x,y ) < epsilon1 } is called the epsilon1-ball about x . Important to note that the definition of openness depends on a metric and a universe . Technically, we should write B d ( x,epsilon1 | X ) Examples: 1 2 LECTURE #4: TUE, SEP 11, 2007 PRINT DATE: AUGUST 21, 2007 (ANALYSIS4) • Suppose X = [0 , 1] and consider the epsilon1-ball about 1: it is (1- epsilon1, 1]. • Suppose d is the discrete metric and X = R . For epsilon1 ≤ 1, the epsilon1-ball about x ∈ R is the point itself. For epsilon1 > 1, the epsilon1-ball about x ∈ R is R . More examples: • Suppose X = R n and d is the max metric, i.e., d ( x , y ) = max {| x i- y i | : i = 1 ,...,n } . What’s the shape of the ball? It’s an n ’dimensional cube. • Suppose X = R n and d is the absolute value metric, i.e., d ( x , y ) = ∑ n i =1 | x i- y i | . What’s the shape of the ball? It’s the n-dimensional version of a diamond. From now on, we will usually take for granted that the universe is R n and the metric is the Pythagorean metric. Unless we specify that these are otherwise, we will simply write B ( x,epsilon1 ). Definition: A set A ⊂ X is said to be open in X w.r.t. a metric d if for every x ∈ A , there exists epsilon1 > 0 such that B d ( x,epsilon1 | X ) ⊂ A ....
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at Berkeley.

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mathAnalysis4-07-draft - P r e l i m i n a r y d r a f t o...

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